Order = 44352000 = 29.32.53.7.11.
Mult = 2.
Out = 2.

## Porting notes

Porting incomplete.

## Standard generators

Standard generators of HS are a, b where a is in class 2A, b is in class 5A and ab has order 11.

Standard generators of 2.HS are preimages A, B where B has order 5 and AB has order 11.

Standard generators of HS:2 are c, d where c is in class 2C, d is in class 5C and cd has order 30.

Standard generators of 2.HS.2 are preimages C, D where D has order 5.

## Black box algorithms

### Finding generators

Group Algorithm File
HS
HS:2

### Checking generators (semi-presentations)

Group Semi-presentation File
HS 〈〈 a, b | o(a) = 2, o(b) = 5, o(ab) = 11, o(ab2) = 10, o(abab2) = 15 〉〉 Download
HS:2 〈〈 c, d | o(c) = 2, o(d) = 5, o(cd) = 30, o([c,d]) = 3 〉〉 Download

## Presentations

HS a, b | a2 = b5 = (ab)11 = (ab2)10 = [a, b]5 = [a, b2]6 = [a, bab]3 = ababab2ab−1ab−2ab−1ab2abab(ab−2)4 = ab(ab2(ab−2)2)2ab2abab2(ab−1ab2)2 = abab(ab2)2ab(ab−1)2ab(ab2)2ababab−2ab−1ab−2 = 1 〉 Details
HS:2 c, d | c2 = d5 = [c, d]3 = [c, d2]4 = ((cd)4cd−2cd−1cd−1cd−2)2 = [c, dcdcd2cd−2(cd2)2cd−1] = [c, dcdcd−2cd−1(cd−2)3] = [c, d−1cd−1cd2cd(cd2)3] = 1 〉 Details

## Representations

### Representations of 2.HS.2

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
1408StdDetails
• Matrix representations
Char Ring Dimension ID Generators Description Link
3GF(9)56StdDetails
3GF(3)112StdDetails
5GF(5)56StdDetails

## Maximal subgroups

### Maximal subgroups of HS

Subgroup Order Index Programs/reps
M22 443 520 100Program: Standard generators
U3(5):2 252 000 176Program: Standard generators
U3(5):2 252 000 176Program: Standard generators
L3(4):21 40 320 1 100Program: Standard generators
S8 40 320 1 100Program: Standard generators
24.S6 11 520 3 850Program: Generators mapping onto standard generators
43:L3(2) 10 752 4 125Program: Generators mapping onto standard generators
M11 7 920 5 600Program: Standard generators
M11 7 920 5 600Program: Standard generators
4.24.S5 7 680 5 775Program: Generators mapping onto standard generators
2 × A6.22 2 880 15 400Program: Generators
5:4 × A5 1 200 36 960Program: Generators mapping onto standard generators

### Maximal subgroups of HS:2

Subgroup Order Index Programs/reps
HS 44 352 000 2Program: Standard generators
M22:2. 887 040 100Program: Generators
Program: Generators
L3(4):22 80 640 1 100Program: Generators
S8 × 2 80 640 1 100Program: Generators
Program: Generators
25.S6 23 040 3 850Program: Generators
43:(2 × L3(2)) 21 504 4 125Program: Generators
Program: Generators
21+6.S5 15 360 5 775Program: Generators
(2 × A6.22).2 5 760 15 400Program: Generators
51+2:[25] 4 000 22 176Program: Generators
5:4 × S5 2 400 36 960Program: Generators

## Conjugacy classes

### Conjugacy classes of HS

Conjugacy class Centraliser order Power up Class rep(s)
1A44 352 000 (ababbb)8
2A7 680 4A 4B 4C 6B 8A 8B 8C 10A 12A 20A 20B (ababbb)4
2B2 880 6A 10B (ababbbabababbabaababbbabababbababbbabababb)3
3A360 6A 6B 12A 15A (ababbbabababbabaababbbabababbababbbabababb)2
4A3 840 12A 20A 20B (ababbbabababbabaababbbabababb)3
4B256 8A (ababbb)2
4C64 8B 8C (abaababbbabababb)2
5A500 10A 20A 20B (aababbbabababb)4
5B300 10B 15A (abb)2
5C25 ababbbabababb
6A36 ababbbabababbabaababbbabababbababbbabababb
6B24 12A (ababbbabababbabaababbbabababb)2
7A7 abababb
8A16 ababbb
8B16 abaababbbabababb
8C16 ababbababbbabababbabaababbbabababbababbbabababb
10A20 20A 20B (aababbbabababb)2
10B20 abb
11A11 11B2 ab
11B11 11A2 (ab)2
12A12 ababbbabababbabaababbbabababb
15A15 ababb
20A20 20B11 aababbbabababb
20B20 20A11 (aababbbabababb)11

### Conjugacy classes of HS:2

Conjugacy class Centraliser order Power up Class rep(s)
1A88 704 000
2A15 360 4A 4B 4C 6B 8A 8B 10A 12A 20A 4D 4E 8C 8D 12B 20C 20D
2B5 760 6A 10B 4F 20B
3A720 6A 6B 12A 15A 6C 6D 6E 12B 30A
4A7 680 12A 20A
4B512 8A 8C 8D
4C128 8B
5A1 000 10A 20A 20C 20D
5B600 10B 15A 10C 20B 30A
5C50 10D
6A72
6B48 12A 12B
7A14 14A
8A32
8B16
10A40 20A 20C 20D
10B40 20B
11A11
12A24
15A30 30A
20A20
2C80 640 6C 6D 10C 14A 30A
2D3 840 6E 10D
4D640 20C 20D
4E192 12B
4F80 20B
6C720 30A
6D72
6E48
8C64
8D64
10C60 30A
10D10
12B24
14A14
20B20
20C20 20D3
20D20 20C3
30A30